 Author
 Carol Zamfirescu (UGent)
 Organization
 Abstract
 Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex‐deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.
 Keywords
 Combinatorics, Graph Theory, Hamiltonian
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU8588698
 Chicago
 Zamfirescu, Carol. 2019. “Cubic Vertices in Planar Hypohamiltonian Graphs.” Journal of Graph Theory 90 (2): 189–207.
 APA
 Zamfirescu, C. (2019). Cubic vertices in planar hypohamiltonian graphs. Journal of Graph Theory, 90(2), 189–207.
 Vancouver
 1.Zamfirescu C. Cubic vertices in planar hypohamiltonian graphs. Journal of Graph Theory. Wiley; 2019;90(2):189–207.
 MLA
 Zamfirescu, Carol. “Cubic Vertices in Planar Hypohamiltonian Graphs.” Journal of Graph Theory 90.2 (2019): 189–207. Print.
@article{8588698, abstract = {Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex\unmatched{2010}deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 6381]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.}, author = {Zamfirescu, Carol}, issn = {03649024}, journal = {Journal of Graph Theory}, language = {eng}, number = {2}, pages = {189207}, publisher = {Wiley}, title = {Cubic vertices in planar hypohamiltonian graphs}, url = {http://dx.doi.org/10.1002/jgt.22388}, volume = {90}, year = {2019}, }
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